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MEASURING A NOVEL
OPTICAL SPRING EFFECT
by
Baylee Danz
A senior thesis submitted to the faculty of
Brigham Young University - Idaho
in partial fulfillment of the requirements for the degree of
Bachelor of Science
Department of Physics
Brigham Young University - Idaho
April 2018
Copyright c© 2018 Baylee Danz
All Rights Reserved
BRIGHAM YOUNG UNIVERSITY - IDAHO
DEPARTMENT APPROVAL
of a senior thesis submitted by
Baylee Danz
This thesis has been reviewed by the research advisor, research coordinator,and department chair and has been found to be satisfactory.
Date Richard Hatt, Advisor
Date Evan Hansen, Comittee Member
Date Jon Johnson, Comittee Member
Date Stephen McNeil, Department Chair
Date R. Todd Lines, Senior Thesis Coordinator
ABSTRACT
MEASURING A NOVEL
OPTICAL SPRING EFFECT
Baylee Danz
Department of Physics
Bachelor of Science
Current gravitational wave detectors employ extremely high laser power levels
in order to reduce the counting error of photons, a type of quantum noise
called shot noise, that arises from quantum mechanics. As a consequence of
this high power, radiation pressure creates a significant opto-mechanical cou-
pling between the laser field and the mechanical motion of the mirrors (also
known as test masses). This optical spring effect can in principle be used to
reduce the quantum noise below the Standard Quantum Limit (SQL). In this
experiment, we aim to learn more about the optical spring effect to broaden
our knowledge of its behavior and possible applications in future gravitational
wave detector advancements. This research focuses specifically on measuring
the optical spring effect in a configuration without an optical cavity. A de-
tailed description of the experimental process of assembling the configuration
is described.
ACKNOWLEDGMENTS
I would like to thank my family for supporting me through all of the chal-
lenges I have faced and for their unyielding faith in me even through my
doubts.
I would like to thank my mentor at Louisiana State University, Dr. Thomas
Corbitt, for accepting me as a research student for this project. I would also
like to thank Jonathan Cripe at Louisiana State University for all the effort
and time he put into helping me, as well as my research group from that REU. I
am incredibly grateful to the National Science Foundation for supporting this
work through the REU Site in Physics and Astronomy (NSF grant number
1560212) at Louisiana State University.
A special thanks to my professors at Brigham Young University Idaho who
have encouraged, guided, and shaped my undergraduate studies; I could not
have succeeded without your help. Thank you to my advisor Brother Hatt, my
other committee members, Brother Hansen, Brother Johnson, Brother McNeil,
and my senior thesis coordinator Brother Lines.
Thank you to all the people who made this possible. I couldn’t have done
it alone.
Contents
Table of Contents xi
List of Figures xiii
1 Introduction 11.1 Gravitational Waves and LIGO . . . . . . . . . . . . . . . . . . . . . 11.2 Noise and the Standard Quantum Limit . . . . . . . . . . . . . . . . 41.3 The Optical Spring Effect . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Design 92.1 Basic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Experiment 153.1 Configuration Distances and Optics . . . . . . . . . . . . . . . . . . . 153.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Locking the Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Results 33
5 Conclusion 35
Bibliography 36
A MatLab Code to Optimize Optics’ Positions 41
xi
List of Figures
1.1 Basic Michelson interferometer used in LIGO. . . . . . . . . . . . . . 31.2 Constructive and destructive interference. . . . . . . . . . . . . . . . 41.3 LIGO quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Version of a Michelson-Sagnac Interferometer that was used as thebasic configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Photograph of a cantilever chip. . . . . . . . . . . . . . . . . . . . . . 112.3 Diagram and photo of the structure of one cantilever mirror. . . . . . 122.4 Representation of the optical fields incident on and resulting from the
cantilever. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Basic Mach-Zehnder configuration. . . . . . . . . . . . . . . . . . . . 163.2 Configuration schematic. . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Photograph of the configuration built as seen from above. . . . . . . . 213.4 Example of a camera image shown from a port at minimum power. . 243.5 Photodetector signals from Port A and the dark port at equivalent
time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Gain vs frequency and phase vs frequency graphs at 50 mW, 100 mW,200 mW, and 360 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
xiii
Chapter 1
Introduction
The detection of gravitational waves has sparked numerous research projects focused
on improving gravitational wave detection technology. This technology is currently
limited by the standard quantum limit, but efforts are being made to develop im-
provements to minimize the quantum noise. This chapter will focus on the reasons
behind investigating the optical spring effect.
1.1 Gravitational Waves and LIGO
The Laser Interferometer Gravitational-Wave Observatory (LIGO) made the first de-
tection in 2015 of a gravitational wave, predicted 100 years prior by Albert Einstein′s
theory of relativity. A gravitational wave is a perturbation in spacetime caused by
a large event involving massive astronomical objects [1]. LIGO has announced six
detected events as of January 2018. Five of the gravitational waves detected have
been the result of black hole mergers. These black holes have ranged from 7 to 36
solar masses. The merging released an enormous amount of energy: the first detected
merger produced waves that radiated 3M�c2. The first frequencies detected were
1
2 Chapter 1 Introduction
35 − 450 Hz. In addition to the five black hole mergers, a gravitational wave was
detected as a result of a binary neutron star system. The most recent announced
detection was a black hole merger in June 2017, announced November 2017 [2–7].
There are currently two LIGO facilities, one in Livingston, Louisiana and the
other in Hanford, Washington. LIGO works closely with the Virgo Collaboration
and European Gravitational Observatory (EGO) in Cascina, Italy, a similar inter-
ferometric facility aimed at detecting gravitational waves [8]. LIGO is based on a
Michelson-Sagnac configuration comprised of a 200-Watt laser split by a beamsplit-
ter along two perpendicular 4-kilometer arms. In order to increase the sensitivity of
the detector, Fabry-Perot cavities with signal recycling mirrors were constructed on
each interferometric arm. These cavities serve two purposes:
• Light is reflected about 280 times within these cavities, effectively increasing the
length of LIGO to magnify any perturbations that are detected. This improves
LIGO sensitivity to gravitational waves.
• Power is magnified within the cavities. Although a 200 W laser is used as the
input, the signal recycling mirrors increase power up to 750 kW. This improves
the resolution of any signals that LIGO detects [9].
1.1 Gravitational Waves and LIGO 3
Figure 1.1 Basic Michelson interferometer used in LIGO. The input laser,seen left, is incident on a beamsplitter and split along two perpendicular 4km arms (indicated by green brackets, up and right). The signal recyclingcavitites are labeled as Fabry Perot cavities, evident by the signal recyclingmirrors facing one another along the arms. The resulting signal from beaminterference is gathered at the dark port, shown by a black dot (bottom) [9].
Through partially reflective mirrors, the high-power light is able to return to the
intersection between the 4 km arms. Without any perturbation, the two beams are
perfectly out of phase and interfere destructively at the dark port. As a gravitational
wave passes over the facility and bends spacetime, the arm lengths are distorted by
a fraction of the size of a proton. This length difference changes the phase of the
beams, and they can interfere constructively at the dark port. These changes in the
4 Chapter 1 Introduction
interference pattern is how LIGO sees gravitational waves.
=
+ +
=
Figure 1.2 Two beams that are in phase interfere constructively, creating apattern with an even taller amplitude (left). If the beams are not in phase andare perfectly out of phase, they interfere destructively, canceling amplitudesout and resulting in no visible signal (right). Adapted from [10].
1.2 Noise and the Standard Quantum Limit
LIGO is limited by various noise sources because of its high sensitivity. Scientists are
aware of these noise sources and the threat they pose by masking gravitational wave
signals, so all known noise sources are taken into account while analyzing signals.
These sources are targeted and removed from the final data using Wiener filtering
and specific noise subtraction analyses [2].
Some of the most prominent noise sources include:
• Seismic vibrations. These limit detection at low frequencies and can include vi-
brations in the Earths crust caused by tectonic movement, human movement,
storms, or ocean waves hitting the shore hundreds of miles away. Human move-
ments that cause noise may include anything from footsteps to nearby traffic.
• Thermal noise. This limits detection at midrange frequencies. Thermal noise
arises from the increased movement of mirrors particles when heated by the
1.2 Noise and the Standard Quantum Limit 5
laser as well as the distortions in materials that the temperature causes. Cooling
systems are in place to minimize this noise.
• Quantum noise. Two types of quantum noise are:
– Radiation pressure noise. This is caused by a quantum uncertainty in the
position of laser photons imparting a radiation pressure force on the mir-
rors. This limits detection at low frequencies.
– Shot noise. This noise is caused by a quantum uncertainty in the position of
photons at output. Shot noise limits detection at high frequencies [11, 12].
Due to the improvements of Advanced LIGO, the main limiting factor of current
and future detection is quantum noise, though steps are still being taken to minimize
the other types of noise.
Due to the Heisenberg Uncertainty Principle in quantum mechanics, the position
and momentum of a particle are impossible to measure simultaneously to high preci-
sion, creating an uncertainty in the prediction of where a laser photon will be. This
principle leads to uncertainty in predicting the behavior of photons interacting with
the test masses. Due to the fluctuations in photons, there is uncertainty in determin-
ing how much and where radiation pressure force will be exerted on the mirrors. The
inability to determine exactly where a mass will be on the quantum level creates the
standard quantum limit (SQL) [12]. An example of a LIGO quantum noise budget
and standard quantum limit is shown in Figure 1.3.
6 Chapter 1 Introduction
Figure 1.3 LIGO quantum noise. Measured noise from LIGO is shown inblue (LHO) and design sensitivity is shown in green (SRD). Calculated quan-tum noise at three different input power levels is also shown (QN). Standardquantum limit (SQL) is shown in red [11].
1.3 The Optical Spring Effect
Projects to improve gravitational wave detecting technology are focusing on various
methods of overcoming the SQL. Two such methods are squeezed light injection and
utilizing an optical spring effect, the latter of which is the focus of this thesis. The
optical spring effect is a result of optomechanical coupling between the test masses
(mirrors) and optical field [12]. The basis of this effect is a linear coupling between
the radiation pressure force that the optical field exerts on the test mass, and the
cavity length. This linear relationship exerts a restoring force on the mirrors with a
calculable spring constant. This effect can be seen when LIGO′s Fabry-Perot signal
1.3 The Optical Spring Effect 7
recycling cavities are detuned from resonance. The cavities are currently controlled
by feedback loops to stay in resonance and prevent the optical spring′s antidamping
force from overwhelming the mechanical damping force [13].
8 Chapter 1 Introduction
Chapter 2
Design
In this chapter we outline the basic schematic of the configuration used in this exper-
iment. A brief discussion of the mathematical theory behind this configuration [13]
will also be discussed.
2.1 Basic Configuration
Previous experiments investigating the optical spring effect have done so using a sig-
nal recycling cavity. However, the optical spring effect can also be created in any
configuration that creates a linear coupling between radiation pressure force and dis-
placement of the mirrors [13]. Our study investigates the optical spring effect in a
configuration without a cavity. We began by using a Mach-Zehnder configuration, a
type of interferometer which consists of two optical fields incident on a beamsplitter
at a 45-degree angle. However, when we proceeded with this configuration we encoun-
tered physical conflicts in the positioning of the beamsplitter mount and the spacing
of the lenses (see Chapter 3 for more details). Because of these conflicts, we decided
to use a variation of the Michelson-Sagnac interferometer, controlling the phase of
9
10 Chapter 2 Design
two laser arms incident on a beamsplitter mirror at 180 degrees. A schematic of the
basic configuration is shown in Figure 2.1.
Laser
Lens 1
Lens 2
Dark Port
Camera & Photodetector
Lens 3
Lens 4
P.T. Mirror B
Cantilever
Mirror A & Piezo
Mirror B
P.T. Mirror A
Port A
Camera & Photodetector
Port B
Camera & Photodetector
Figure 2.1 Version of a Michelson-Sagnac Interferometer that was used asthe basic configuration.
The cantilever is a beamsplitter made of layers of gallium arsenide. It is a partially
transmissive mirror. Similar cantilevers were used by Cole, et al. (2008), and the
gallium arsenide layers are candidates for future LIGO test mass coatings [14]. In
our configuration, a cantilever chip was placed at the intersection between the two
optical fields. This cantilever chip contains rows of cantilevers of varying mechanical
properties and sizes, as shown in the photographs below.
The specific cantilever we chose to utilize in our configuration was highly reflective
mirror about 100 microns wide with a mechanical frequency of about 800 Hz. The
2.1 Basic Configuration 11
Figure 2.2 (Top) Photograph of a cantilever chip. Five rows of individualmirrors are visible. (Bottom) Photograph of three different mirrors in oneof the rows of the cantilever chip. Each have different mechanical properties[15].
12 Chapter 2 Design
structure of a cantilever is shown in the diagram below:
Figure 2.3 Diagram and photo of the structure of one cantilever mirror [16].
Our optical configuration utilizes a laser of 1064 nm with maximum power of
500 mW. Power input is controlled by a waveplate not shown in the configuration
diagram, allowing us to record the optical spring effect at varying power levels. Lenses
were used along the laser beams path to focus the beam diameter to a target width
of 20 microns, or at least within the 100 µm-wide cantilever. The specific distances
and properties of the lenses and mirrors used in this configuration will be discussed
in the next chapter.
After being focused by Lenses 1 and 2, the input beam is incident on a beam-
splitter. This initial beamsplitter creates the two optical fields, beams A and B, that
are incident on opposite sides of the cantilever after being directed by mirrors and
focused through Lenses 3 and 4. The transmitted and reflected beams from the can-
tilever exit the configuration at three ports, as shown in the diagram: Port A, Port
B, and the dark port. Beamsplitters were used to analyze the signal from each of
these ports with a camera and a photodetector. Filters were used (not shown in the
2.2 Mathematical Model 13
diagram) when reducing the amount of power incident on a camera or photodetector
was necessary.
In order to control the phase difference of beams incident on the cantilever neces-
sary to create an optical spring effect, a piezo control device was attached to Mirror
A, which changes the length of beam A at a controlled frequency. A one-way filter
was also placed in front of the power input to ensure that reflected beams would not
damage the laser source (not shown in diagram). Specific alignment and procedures
will be discussed in greater detail in the next chapter.
2.2 Mathematical Model
We first define the individual optical fields interacting with the cantilever. To illus-
trate, consider Figure 2.4. Fields a and d are incident on the mirror from opposite
directions. Fields b and c are the combined transmitted beams through the cantilever
and reflected beams from the cantilever from fields d and a.
a c
b d
Figure 2.4 Representation of the optical fields incident on and resultingfrom the cantilever. Fields a and d are from incident beams a and b of theconfiguration. Fields b and c are a combination of the reflected and partiallytransmitted incident beams. Adapted from [13].
The normalized fields can then be mathematically defined as:
a =
√P0
2
b = ρa+ τd
14 Chapter 2 Design
c = τa− ρd
d =
√P0
2eiφ
φ =Lω0
c
with a − d representing the incident and resulting beams, L and φ representing the
length and phase difference between a and d, and ρ and τ representing the reflectivity
and transmissivity constants of the cantilever with the sum of their squares equal to
1 [13]. The net force on the cantilever is determined through the difference between
incident and resulting powers, or Pnet of the cantilever system:
Pnet = |b|2 − |c|2 = 2ρτP0 cosφ
Fnet = (Pa + Pb − Pc − Pd)/c
where P0 is the initial power input [13]. The radiation pressure force FRP can be
gathered from Fnet: if incident fields a and d are balanced and Pb – Pc is a nonzero
value, the radiation pressure force is Pnet/c. This force displaces the cantilever by
some small amount δL around an equilibrium position, creating a differential force
and spring constant KOS according to the equations:
δFRP =1
c
dPnetdL
δL
KOS = −1
c
dPnetdL
=2
c2ωP0ρτ sinφ
For a more detailed description of the optical spring mathematical model for this
configuration, see [13].
Chapter 3
Experiment
This chapter includes specific lengths and properties of the configuration used. We
will also explain the process followed to build the configuration and align the op-
tics. Various problems were encountered throughout the experimental process; these
problems were described along with their respective solutions.
3.1 Configuration Distances and Optics
As mentioned in Chapter 2, we initially began to build a Mach-Zehnder configuration
illustrated in Figure 3.1. However, we encountered complications due to the 45-degree
angle of incidence on the cantilever. At that angle, the mount holding the cantilever
chip would impede the beam’s access to the cantilever mirror. Because the mount
prevented a clear path, we were forced to reconsider what type of configuration to
build. Instead we used the version of a Michelson-Sagnac interferometer introduced
in Chapter 2 that allowed an unimpeded normal incidence onto the cantilever.
15
16 Chapter 3 Experiment
Figure 3.1 Basic Mach-Zehnder configuration [17]. This configuration wouldhave been used had the cantilever mount not interfered with the beam pathsincident onto the cantilever. The cantilever would have been placed at theintersection of red and blue beams closest to the output signals.
Before we began to physically set up the configuration, we needed a blueprint
to follow. After drawing a rough schematic to determine how many optics would
be necessary, we used a computer program to optimize lens positions. The program
took inputs of path length, number of lenses and a range of position that they could
be placed within, and the final desired beam width which was about 20 microns. It
output the positions along the path length of the lenses that would minimize the
beam to the desired width. This MatLab code is shown in Appendix A.
This program provided a viable basis for the configuration but failed to take into
account impossibilities in converting the computational model to a physical model.
For instance, sometimes the best theoretical placement of lenses was within one cen-
timeter of one another. This would prove impossible to recreate on an optics table,
as the thickness of the lenses and the lens mounts prevented such positions. The
program also didn′t take into account the properties of right triangles, such as the
right triangle in part of our configuration. Thus, various parameters had to be put
3.1 Configuration Distances and Optics 17
into the program to ensure the configuration was physically possible.
An optimized diagram of the configuration with optics’ distances in meters from
the origin is shown in Figure 3.2. The flip mirror marks the beginning of the con-
figuration path at 0.0 m, and the cantilever marks the end of the path at 0.636 m.
Also included in this diagram is the laser source. Although this experiment could be
carried out with the laser source directly input into the configuration, due to limited
table availability we diverted our laser from another configuration. The optics from
the other configuration are unnecessary and are not shown.
18 Chapter 3 Experiment
Figure 3.2 a) Configuration schematic. b) Configuration schematic labeledwith optics’ distances (in meters) from the origin at 0.0.
3.1 Configuration Distances and Optics 19
The following is a list of the optics and their properties that were used, beginning
from the laser and following the optical path to the cantilever:
• The laser has a maximum power of 500 mW. The input power to our configu-
ration was controlled by a half waveplate paired with a polarizing beamsplitter.
We adjusted the waveplate for varying power levels. 50 mW were initially used
while setting up our configuration, then varying power levels were recorded once
the configuration was operational and the optical spring effect was noticeable.
• The isolator was used to ensure that light travels only in one direction. If any
beams reflected off of optics and traveled back along the beam path, the laser
source would be damaged.
• The waveplate, paired with a polarizing beamsplitter, was used to change the
input power level by diverting certain polarization components of the laser.
• The following mirror and flip mirror were only in place to divert the laser from
an adjacent project into our configuration. A flip mirror was used to allow easy
laser access to the other configuration if needed.
• The polarizing beamsplitter (PBS) was used to decrease the power input. Spe-
cific polarization was unnecessary to this project, but the PBS diverted certain
polarizations from the beam path, thus diminishing the power input moving
forward. Beam blocks were placed on either side of the PBS to ensure that the
diverted polarized beams didnt interfere with other optics.
• Lens 1 has a focal length of 0.05 m.
• Lens 2 has a focal length of 0.0254 m.
20 Chapter 3 Experiment
• The following optic is a beamsplitter. All of the following beamsplitters are
50% reflective and 50% transmissive (50/50) according to their labels. However,
they were measured to be closer to 42% reflective and 45% transmissive. This
beamsplitter creates beams A and B which converge at the cantilever but also
allows a signal to exit the configuration at the dark port.
• The beamsplitter at the dark port is a 50/50 beamsplitter. It divides the signal
so that the dark port camera and photodetector can read it.
• Mirror B is highly reflective. It has a reflectivity value R > 99%.
• Mirror A also has a reflectivity value R > 99%. This mirror is smaller and
contains the piezoelectric device. The device is glued to the back of the mirror.
When a voltage is sent through the piezo at a certain frequency, it expands
and contracts, allowing control over the phase of beam A through minute path
length changes.
• Port (P.T.) mirror A is 94% reflective, allowing a signal to reach Port A camera
and photodetector.
• Port mirror B is 94% reflective, allowing a signal to reach Port B camera and
photodetector.
• The beamsplitter at Port A is a 50/50 beamsplitter. It divides the signal so
that the Port A camera and photodetector can read it.
• The beamsplitter at Port B is a 50/50 beamsplitter. It divides the signal so
that the Port B camera and photodetector can read it.
• Lens 3 has a focal length of 0.05 m.
• Lens 4 has a focal length of 0.05 m.
3.2 Alignment 21
• The cantilever is 65% reflective. It is made of gallium arsenide and has a
mechanical resonance of about 800 Hz.
Figure 3.3 Photograph of the configuration built as seen from above. Laserbeam paths are indicated by red lines. Ports are indicated by yellow arrows.
3.2 Alignment
The alignment of the configuration required the most attention of any step in this
research project. It consisted of five parts:
1. Optimizing the positions of optics during the initial setup, excluding the can-
tilever.
2. Aligning the beams as much as possible without the cantilever.
3. Setting up the cantilever.
4. Aligning beams individually onto cantilever.
5. Aligning both beams at the same time onto the cantilever.
Throughout each of these steps, measurements of optics positions and signals from
22 Chapter 3 Experiment
the ports were recorded for reference. Recording the configuration positions at each
step was vital; if a mistake was made in subsequent positioning, we could refer to the
previous positioning and retrace our steps to start anew.
Step 1. Initially placing the optics in positions corresponding to the theoretical
model obtained from the computer program proved more difficult than expected.
Once we decided to redo our configuration, as mentioned in Chapter 2, to the version
of the Michelson-Sagnac interferometer shown in the diagram, we had to take down
our previous work and start anew. Confined by limited table space, the computer
model positions, and human error, we had to retrace our steps a few times to ensure
the optics would be placed correctly.
We used a methodical approach of setting up one optic at a time, beginning closest
to the laser and working towards where the cantilever will be placed. While securing
the optics on the table, the laser was blocked for safety purposes. It was only used
to make sure the optics were placed properly: when placing a lens, we adjusted it
so that the beam passed through its center as much as we could determine with a
detector card. We then used a ruler to make sure that the resulting beam wasn′t
tilted horizontally or vertically by measuring its position at a close and far distance
and adjusting the lens accordingly. The goal was to make sure the beam was parallel
to a plane 3 inches above the table throughout the configuration, and perpendicular
to all of the optics. Placing beamsplitters was similar except we had to check that
the paths of both resulting beams were straight. When we optimized the positions of
the mirrors, we centered the beam on the mirror and used gridlines on the table to
ensure that the reflected beam was straight along the desired path.
Lenses 3 and 4 were initially on stationary lens mounts that provided minute
adjustments. These lenses′ positions are vital in focusing the beam width down to
20 microns before interacting with the cantilever. We were incapable of placing the
3.2 Alignment 23
optics in the exact position that the mathematical model suggested due to imperfect
estimations of distance. We were able to place the optics within about a centimeter
of the mathematically perfect position, but a range of error existed. Because of this,
we needed to be able to adjust the lenses positions along the path, and their current
mounts did not allow for this adjustment. We needed more dynamic lens mounts after
optimizing beam and optics alignment, but we knew that taking down parts of an
already good configuration to rebuild them in better ways would be more beneficial
to the experiment in the long run. The stationary lens mounts were replaced with
mounts that could move the lens along the path length towards or away from the
cantilever. These adjustable mounts proved vital to our eventual success.
Step 2. Cameras and photodetectors were placed at each of the ports. Initially,
due to limited resources, we placed cameras at two ports and a photodetector at Port
B. However, having both a camera and photodetector at each port proved necessary
to compare ports simultaneously, so we acquired both devices for each port. Cameras
proved most useful in the initial setup: two beams were visible at the ports if beams
A and B were not aligned properly. Optimizing various lens and mirror positions
showed the beams converging on the camera′s image.
The initial configuration did not contain the cantilever due to the delicacy of its
structure and placement. Because the cantilever is small and the beam width was not
yet minimized to 20 microns, we didnt want to risk placing the cantilever in the path
of the beam and breaking one of its mirrors with an uncontrolled radiation pressure
force. All correct beam alignments and correct optics placements were necessary
before placing the cantilever into the configuration.
After aligning the optics and beam path, we placed a beam splitter plate in place
of where the cantilever would be to mimic the cantilever′s partially reflective and par-
tially transmissive behavior. This allowed us to use the ports as tools for optimizing
24 Chapter 3 Experiment
alignment; given that the configuration now matched our mathematical model, the
dark port should be dark if the beam paths were equal. We chose a couple different
optics, namely the beamsplitter that creates beams A and B and arm mirror B, to
optimize. Aligning the beams by moving the optics by hand would be impossible due
to the extreme sensitivity of placement. Instead, we used the small knobs on the optic
mounts to adjust minute placements. At this point the camera at the dark port did
not provide the accuracy we needed in aligning beams, so we used the photodetector.
By slowly adjusting the optics knobs, the power signal shown from the photodetector
increased or decreased. Adjusting two or more optics showed greater minimization of
the signal at the dark port. Our mathematical model predicted that Ports A and B
should be in phase and that the dark port should be shifted in phase by π from each
of these ports. If the dark port was at minimum power, then Ports A and B should
be at maximum power. On this assumption we used the photodetectors to estimate
positions that created a maximum signal at Ports A and B and a minimum signal at
the dark port.
Figure 3.4 Example of a camera image shown from a port at minimumpower.
3.2 Alignment 25
Before placing the cantilever in the configuration, we also tested the piezo control
device to make sure that we could see a change in frequency when the device was
turned on. Using a PID controller, we sent a voltage through the device with a certain
frequency and were able to see that frequency through the photodetector signals at
the ports. This confirmed to us that our configuration was aligned and that we were
able to control the length of beam A.
Figure 3.5 Photodetector signals from Port A and the dark port at equiva-lent time steps. The yellow wave is from the dark port and the red wave isfrom Port A. The two ports are π out of phase: when the yellow signal is ata trough, the red signal is at a peak, and vice versa. The port signals showdifferent powers due to filters placed in front of the dark port detector.
Step 3. Placing the cantilever into the setup proved to be a difficult thought
process because the cantilever chip was small and of unique shape. The chip had
to be able to move in all directions to adjust the very small beam to the very small
mirror, and the mirrors within the chip could not be impeded by any type of mount.
In addition to these limitations, the small space that was left to us between lens 3 and
4 mounts would make a tight squeeze for any cantilever mount we chose. One original
idea to mount the chip was to glue it onto a long focal-length lens. This would provide
26 Chapter 3 Experiment
a nearly flat clear surface that would not impede the beams path to the mirrors,
and the lens mounts were small and dynamic. However, due to a limited number
of cantilever chips, we were reluctant to use such a permanent method as gluing.
Instead, we used an optics mount that could be adjusted in all directions. It was a
large mount, however, and more adjustments had to be made to the configuration in
order to fit the mount in the given space. Once these adjustments were made and
the beams were realigned, the mount proved vital in the final adjustments.
Because the chip contained several rows of mirrors, we chose to focus on a larger
highly reflective mirror so that the reflected beam would be easier to see from the port
cameras. A bright reflection alerted us to when the beam was focused on a mirror
or not. This was not the mirror from which we collected optical spring data but it
helped us refine our configuration before focusing on the final mirror.
One of the initial tasks after mounting the cantilever was to determine if the mirror
we selected was at all bent or tilted. If the mirror′s surface was not perpendicular
to the beam path, the reflected beam would veer to the side or vertically after some
distance. Each cantilever mirror had the potential to be tilted a different way and
affect the reflected beam differently, so we had to choose one mirror to focus on while
adjusting the mount. If a reflected beam was not aligned with the transmitted beam,
we adjusted the pitch and yaw of the cantilever chip until the beams aligned. We
would use the port cameras and photodetectors to make sure the beams were visibly
aligned and that we were receiving the reflected beams′ signals in all ports. To ensure
that the signals from the ports were both transmitted and reflected beams, we would
block one beam at a time and observe the power drop shown by the photodetectors.
Adjusting the pitch and yaw of the cantilever chip was a process that would need to
be repeated once the beam width was minimized and focused onto the final mirror.
Step 4. To align the beams onto the cantilever, we focused on one beam at a
3.2 Alignment 27
time, blocking beam A to adjust beam B′s position and blocking beam B to adjust
beam A′s position. When a beam is blocked, transmitted light is seen from one port
and reflected light on the other. When beam A is blocked, transmitted light is seen
at Port A and reflected light is seen at Port B and the dark port. This knowledge
of what should be seen at each port was vital while adjusting the beam widths and
position on the cantilever.
The most important step at this point was to minimize the beam width to less
than the dimensions of the cantilever mirror. We had to interpret what was seen from
the cameras and photodetectors to determine the width of the beam.
If the beam was significantly bigger than the mirror:
• A shadow could be seen from the port cameras that showed the outline of each
mirror or the rows between mirrors.
• Transmitted light would be visible at the port that is supposed to detect only
reflected light. Reflected light would be minimal.
• Depending on how much wider the beam was than the mirror, a circular in-
terference pattern could be seen from transmitted light interfering with itself
around the circular mirror. This type of pattern was seen in the beginning of
our cantilever adjustments.
If the beam was smaller than the mirror:
• Reflected light would be obvious at one port and at the dark port. At first
the reflected beam was not bright enough to detect from port cameras but
was significant enough to detect on photodetectors. Eventually we were able
to clearly see the reflected beam at the respective port once the camera and
cantilever were adjusted.
28 Chapter 3 Experiment
• Transmitted light would not be visible at the respective port when covering one
beam.
Minimizing beam size was the first step. Determining onto which cantilever mirror
a beam was focused was the next step. The only way to determine onto which mirror a
beam was focused was to move the cantilever chip and observe the behavior of reflected
and transmitted beams when scanning across the chip (refer to cantilever image in
Chapter 2). Relying heavily on the images of reflected and transmitted beams that
the port cameras provided us, we used the vertical and horizontal adjustment knobs
on the cantilever mount to determine our placement on the chip.
Turning the horizontal adjustment knob scanned across the chip row. If the beam
was originally focused on a mirror, we would see the reflected beam disappear and
transmitted beam appear when focusing in the space left or right of the mirror.
Continuing to scan left or right would bring the beam into contact with other mirrors
on the same row or the edge of the chip. Other mirrors were evidenced by alternating
reflected and transmitted beams, and the edge of the chip was seen as a large distorted
reflected image and no transmitted signal.
Turning the vertical adjustment knob scanned across multiple rows and the chip
edge between rows. If the beam was originally focused on the mirror, the reflected
beam disappeared and the transmitted beam appeared when focusing above or below
the mirror. The transmitted beam would then disappear and the reflected beam would
appear distorted when the beam focused on the chip edge between mirror rows. A
transmitted beam would reappear when the beam focused on the row above or below
our original mirror. If another cantilever mirror was on the row directly above or
below our original mirror, scanning vertically would show alternating transmitted
and reflected beams depending on whether the beam came into contact with the chip
edge or a mirror.
3.2 Alignment 29
Carefully observing the reflected and transmitted images while moving the can-
tilever chip determined onto which area of the chip the beam was focused. We indi-
vidually determined onto which mirror a beam was focused by repeating this process
while blocking one beam at a time.
The signal from the ports also confirmed that we were focused on a mirror when
we could see the image or the photodetector signal oscillating when the table was
bumped or a loud noise was made. This did not confirm onto which mirror the beam
was focused, but it did confirm that the beam was focused on a mirror. The sensitivity
of the cantilever mirrors to any type of vibration was evidenced by the photodetector
signal oscillating or the camera image vibrating slightly.
Step 5. Once we were fairly certain that the beams were focused on the same
mirror, we unblocked both beams. If both were focused on the same mirror, an
interference pattern would be visible at all the ports. The interference pattern was
a result of the transmitted beam through the cantilever and the reflected beam from
the cantilever, from both beams A and B.
When we initially unblocked both beams, a distinct interference pattern was not
visible. We also saw that only part of the camera image oscillated when a loud noise
was introduced to the cantilever. We eventually determined that despite our previous
efforts, beams A and B were not focused on the same cantilever mirror, and we had
to repeat the previous alignment process.
In time both beams focused on the same mirror and an interference pattern was
visible from all three ports. Despite both beams being focused on the same mirror,
they were not perfectly overlapping. More minor adjustments were made to various
optics in order to minimize the dark port and investigate the interference patterns
shown from the cameras. The final adjustments to minimize the dark port signal and
maximize Port A and B signals consisted of realigning various optics. Optimizing two
30 Chapter 3 Experiment
or more optics at once, lenses and mirrors and beamsplitter included, was necessary
to perfect the signals.
Miscellaneous problems encountered:
• Throughout this process we happened upon various stray beams whose source
we could not initially determine. Due to safety precautions, we set up beam
blocks whenever we found one of these stray beams veering from the configura-
tion. We determined that many of these beams were from reflections of optics
coatings.
• We also detected a strange interference pattern for transmitted light from the
cantilever, but only when one of the beams was blocked. We could not determine
the source but believe it was from an impurity or fingerprint on one of the optics
surfaces.
• Transmitted and reflected light should be inversely correlated, but they weren′t
initially. This was due to improper alignment of beams on one mirror. With
more readjustments, an inverse relationship could be seen between the dark
port and Ports A and B.
• Theoretically, Port A and B signals should be identical. However, they were
not. Some signals were more powerful than others, and some beams seen from
the port cameras had different shapes than the other ports. This could be due
to a number of factors, namely, human errors in the optics placements. This
could also be due the fact that the distances between P.T. Mirror 2 and the
Port A and B detection devices were not equal. The beams were distorted or
faded over some distance, and due to the confinement of our table, we did not
place the cameras and photodetectors at equal distances from their respective
port mirrors.
3.3 Locking the Interferometer 31
• Sometimes signals reaching the ports were stronger or weaker than expected. If
stronger, filters were placed in front of the cameras or photodetectors to prevent
damage to the devices. If weaker, settings on the cameras or photodetectors
needed to be adjusted to detect the low power.
• Optics would misalign easily due to the sensitivity of placement. An accidental
nudge to a lens mount or turning a knob a little bit too far would completely
misalign the beams. Each of the steps in the alignment process needed many
readjustments when beams would misalign.
• The pitch and yaw of the cantilever had to be readjusted when we chose the
final mirror onto which we would focus the beams. The initial mirror onto which
we focused was large and highly reflective; the final mirror was slightly smaller
and also highly reflective. We chose this mirror because we believed that we
would be able to see the optical spring effect more clearly due to its mechanical
properties.
3.3 Locking the Interferometer
The cantilever mirrors were extremely sensitive to noise produced by physical distur-
bances to the optics table or loud noises. When the configuration was aligned onto
the desired mirror, oscillations were visible from the photodetectors when the table
was bumped, when people spoke near the configuration, or when loud devices were
turned on elsewhere in the same room. Putting the configuration in a vacuum would
reduce noise and make the optical spring effect more visible. Due to limited time,
space, and resources, we did not use a vacuum to gather initial optical spring data.
However, future improvements of this project may include using a vacuum chamber
to minimize noise.
32 Chapter 3 Experiment
To begin taking data and recording the dark port signal, the configuration had
to be locked. This meant using the piezo device to counteract oscillations caused
by noise and locking the configuration at a specific frequency. We connected the
piezo and the dark port signal through a feedback loop using a PID controller and
an integrator. When noise produced an oscillation at the dark port photodetector
different than the set frequency, the feedback loop would calculate the difference and
signal to the piezo to counteract the noise. The phase difference between beams A
and B is controlled by setting the lock point on the PID controller. The feedback
loop is also how we measure the optical spring effect. By introducing a signal to the
loop, we measure the response of the system to the controlled signal, which will show
the spring effect.
Chapter 4
Results
The response of the dark port photodetector to an excitation on the piezo was mea-
sured in the frequency domain at varying input power levels controlled by the wave-
plate: 50 mW, 100 mW, 200 mW, and 360 mW. The results are shown in Figure
4.1.
Figure 4.1 Gain vs frequency and phase vs frequency graphs at 50 mW, 100mW, 200 mW, and 360 mW. Mathematical predictions are shown by dashedlines and measured values are shown by solid lines.
33
34 Chapter 4 Results
The amplitude graphs dip at the mechanical resonance around 800 Hz. The optical
spring frequency can be seen at 4.2 kHz. The theoretical model was created by other
members of this research group. The experimental data and theoretical model show
similar patterns; the mathematical prediction is shown on this graph as a dotted line.
The data also showed that measurements taken at higher powers produce a larger
and clearer optical spring effect.
Chapter 5
Conclusion
We effectively measured the optical spring effect at four different power levels by
measuring the response of the system to an introduced signal. Our results matched our
theoretical models. The optical spring effect increases with power input. The piezo
control and feedback loop ensured the stability of the configuration by controlling the
phase difference between interferometer arms and suppressing noise.
Future applications of this research may include:
• Using a signal recycling cavity to increase the laser power in order to see a
greater optical spring effect.
• Increasing power levels to increase noise suppression.
• Putting the experiment in a vacuum to minimize disturbances caused by air
molecules and increase noise suppression.
• Investigate thermal noise of the system.
• Investigate properties of cantilevers, which may be used in future interferometric
technology.
35
36 Chapter 5 Conclusion
• Investigate more thoroughly how the optical spring effect can be used to reduce
noise below the SQL in LIGO advancements.
Although we successfully detected the optical spring effect in this configuration,
this research has not seen its end. Our configuration may continue to be used in
future projects to more clearly understand how the optical spring effect can improve
interferometric gravitational wave detection below the SQL.
Bibliography
[1] LIGO, “What are gravitational waves?.” https://www.ligo.caltech.edu/page/
what-are-gw. Accessed: 2018.
[2] B. P. Abbott et al., “Observation of gravitational waves from a binary black hole
merger,” Phys. Rev. Lett., vol. 116, p. 061102, Feb 2016.
[3] B. P. Abbott et al., “Gw151226: Observation of gravitational waves from a 22-
solar-mass binary black hole coalescence,” Phys. Rev. Lett., vol. 116, p. 241103,
Jun 2016.
[4] B. P. Abbott et al., “Gw170104: Observation of a 50-solar-mass binary black
hole coalescence at redshift 0.2,” Phys. Rev. Lett., vol. 118, p. 221101, Jun 2017.
[5] B. P. Abbott et al., “Gw170608: Observation of a 19 solar-mass binary black hole
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[6] B. P. Abbott et al., “Gw170814: A three-detector observation of gravitational
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[7] B. P. Abbott et al., “Gw170817: Observation of gravitational waves from a binary
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37
38 BIBLIOGRAPHY
[8] G. Gonzalez et al., “Memorandum of understanding between VIRGO on one side
and the laser interferometer gravitational wave observatory (LIGO) on the other
side,” LIGO Document M060038-v2, 2014.
[9] LIGO, “Ligo’s interferometer.” https://www.ligo.caltech.edu/page/ligos-ifo. Ac-
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[10] LIGO, “What is an interferometer.” https://www.ligo.caltech.edu/page/what-
is-interferometer. Accessed: 2018.
[11] T. Corbitt, Quantum Noise and Radiation Pressure Effects in High Power Op-
tical Interferometers. PhD thesis, Massachusetts Institute of Technology, Cam-
bridge, Massachusetts, 2008.
[12] T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferom-
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[13] J. Cripe, B. Danz, B. Lane, M. Catherine Lorio, J. Falcone, G. Cole, and T. Cor-
bitt, “Observation of an optical spring with a beamsplitter,” 12 2017.
[14] G. Cole, S. Groblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrys-
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[15] A. Libson and T. Corbitt, “Ponderomotive squeezing and optomechanics.” Grav-
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BIBLIOGRAPHY 39
[17] QuantumMoxie, “A simple but definitive guide to Mach-Zehnder inter-
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40 BIBLIOGRAPHY
Appendix A
MatLab Code to Optimize Optics’
Positions
%\begin {verbatum}
% −−−−−−−−−− Scr ipy us ing a l a mode mode matching u t i l i t i e s −−−−−−−−−−−−
% Fi r s t attempt at mode matching f o r new cav i ty
%Al l va lue s in m
c l o s e a l l
c l e a r c l a s s e s
% c r ea t e a new beam path ob j e c t
CAVITYypath = beamPath ;
% add components to the beam path
CAVITYypath . addComponent ( component . l e n s ( 0 . 05 , . 1 4 , ’ Lens1 ’ ) )
CAVITYypath . addComponent ( component . l e n s (0 .0254 , . 4826 , ’ Lens2 ’ ) )
CAVITYypath . addComponent ( component . l e n s ( 0 . 05 , . 4826 , ’ Lens3 ’ ) )
% l en s syntax : ( f o c a l length , z po s i t i on , s t r i n g l a b e l )
%CAVITYypath . addComponent ( component . l e n s ( . 05 , − . 03 , ’ SL2 ’ ) ) ;
%CAVITYypath . addComponent ( component . l e n s ( . 05 , − . 3 , ’ SLnew ’ ) ) ;
%Ste e r i ng mi r ro r s a f t e r the f i b e r output
CAVITYypath . addComponent ( component . f l a tM i r r o r ( 0 , ’ f l i pM i r r o r ’ ) )
CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 2 5 , ’BS ’ ) )
CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 3 171 , ’ armMirror1 ’ ) )
41
42 Chapter A MatLab Code to Optimize Optics’ Positions
CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 3 848 , ’ beamsCross ’ ) )
CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 5 317 , ’ armMirror2 ’ ) )
CAVITYypath . addComponent ( component . f l a tM i r r o r ( . 6 3 6 , ’ c an t i l e v e r ’ ) )
% f l a t mirror : ( z po s i t i on , s t r i n g l a b e l )
% f l a t mi r ro r s don ’ t change modematching but they l e t you know i f you ’ re going to be putt ing
% s t u f f on top o f eachother .
% The other u s e f u l component part i s curved mirror , to make a curved mirror :
% component . curvedMirror ( rad iu s o f curvature , z po s i t i on , l a b e l )
%CAVITYypath . addComponent ( component . l e n s ( .025 , − . 015 , ’ Glued lens2 ’ ) ) ;
% de f i n e ” input beam” but i t doesn ’ t have to be at the input , i t can be anywhere in the beam path
CAVITYypath . seedWaist (124 . 9 e−6 ,− .24);
% seedWaist syntax : ( wais t width , z p o s i t i o n )
% de f i n e the beam you are t ry ing to match into , the t a r g e t .
CAVITYypath . targetWaist (20 e−6, . 6 3 6 ) ;
% targetWaist syntax : ( wais t width , z p o s i t i o n )
% s l i d e components to opt imize mode over lap .
%CAVITYypath = CAVITYypath . optimizePath ( ’ Col l imator1 ’ , [−1 − . 9 2 ] , . . .
% ’ Col l imator2 ’ , [ − . 88 − . 72 ] , ’ SL1 ’ , [ − . 2 − . 07 ] , ’ SL2 ’ , [ − . 1 − . 03 ] ) ;
%CAVITYypath = CAVITYypath . optimizePath ( ’ Glued lens ’ , [− .07 − .03 ] , ’ Outside1 ’ , [− .95 − . 7 ] , . . .
% ’ Outside2 ’ , [− .85 − .65 ])
CAVITYypath = CAVITYypath . optimizePath ( ’ Lens1 ’ , [ . 1 4 . 2 4 ] , ’ Lens3 ’ , [ . 3 7 . 6 ] , ’ Lens2 ’ , [ . 1 5 . 2 3 ] )
% optimizePath syntax ( component name , [ ( lower bound ) ( upper bound ) ] , another component name , . . . )
% you can choose to opt imize as many components as you would wish . Result i s s e n s i t i v e to i n i t i a l
% cond i t i on s which are de f ined by the z po s i t i o n o f the components be f o r e running opt imize path .
% You can make i t unbounded on e i t h e r or both s i d e s by us ing i n f .
% i f a component i s not named , i t w i l l s tay put .
%% a f t e r y path opt imized
% dup l i c a t e the opt imized beampath in order to work with the components e x c l u s i v e to the x path
CAVITYxpath = CAVITYypath . dup l i c a t e ;
% I f you j u s t did PSLxpath = PSLypath ; you would j u s t have two names f o r the same object ,
43
% changing one would change the other . ( th ink po i n t e r s )
%the x path has a d i f f e r e n t s t a r t i n g wais t than the y path
CAVITYxpath . seedWaist (124 . 9 e−6 ,− .24);
% add a c y l i n d r i c a l l e n s
%CAVITYxpath . addComponent ( component . l e n s ( . 7 5 2 , . 3 , ’CL1 ’ ) ) ;
% opt imize the po s i t i o n o f the c y l i n d r i c a l l e n s
%CAVITYxpath = CAVITYxpath . optimizePath ( ’CL1 ’ , [ . 2 5 . 2 9 ] ) ;
% the targetOver lap method c a l c u l a t e s the mode over lap assuming i t ’ s doing an
% x and y i n t e g r a l , because these are a c t ua l l y the two dimensions o f the same
% beam we square root and mult ip ly them toge the r .
modematch = sq r t (CAVITYxpath . targetOver lap ∗CAVITYypath . targetOver lap ) ;
d i sp ( [ ’ modematching = ’ , num2str (modematch ) ] )
%% plo t
% de f i n e p l o t t i n g domain
zdomain = − . 3 : . 0 0 1 : . 7 ;
f i g u r e (1 )
subplot ( 2 , 1 , 1 )
hold on % r i gh t now a l l the p l o t commands act l i k e the matlab p l o t command and w i l l
% overwr i t e the e x i s t i n g f i g u r e un l e s s you turn hold on
% The p lo t commands a c t ua l l y p l o t s two t race s , the top and bottom of the beam .
% The output o f the p l o t commands r e tu rn s the p l o t handle o f the top so when we make
% the legend we don ’ t have to put a l a b e l on the top and bottom of the beam .
yp lot = CAVITYypath . plotBeamWidth ( zdomain , ’ b ’ ) ;
xp lo t = CAVITYxpath . plotBeamWidth ( zdomain , ’ r ’ ) ;
CAVITYxpath . plotComponents ( zdomain , 0 , ’ r ∗ ’ ) ;
a x i s t i g h t
legend ( [ yp lo t xp lot ] , ’Y’ , ’X’ ) % i f we didn ’ t use handles we would need to do
44 Chapter A MatLab Code to Optimize Optics’ Positions
% legend ( ’Y top ’ , ’Y bottom ’ , ’X top ’ , ’X bottom ’ ) or something .
y l ab e l ( ’Beam width (m) ’ )
g r i d on
hold o f f
subplot ( 2 , 1 , 2 )
hold on
CAVITYypath . plotGouyPhase ( zdomain , ’ wrap ’ , ’ b ’ ) ;
CAVITYxpath . plotGouyPhase ( zdomain , ’ wrap ’ , ’ r ’ ) ;
CAVITYxpath . plotComponents ( zdomain , 0 , ’ r ∗ ’ ) ;
a x i s t i g h t
g r id on
hold o f f
y l ab e l ( ’Gouy Phase ( degree s ) ’ )
x l ab e l ( ’ a x i a l d i s t ance from MOPA aperture (m) ’ )
s e t ( f i n d a l l ( gcf , ’ type ’ , ’ axes ’ ) , ’ f o n t s i z e ’ , 2 0 )
s e t ( f i n d a l l ( gcf , ’ type ’ , ’ text ’ ) , ’ f on tS i z e ’ , 2 0 )
}